Harmonic (Quantum) Neural Networks

TL;DR

This paper introduces methods to incorporate harmonic function biases into neural networks, including quantum neural networks, and demonstrates their effectiveness through benchmarking against physics-informed models.

Contribution

It presents novel techniques for representing harmonic functions in neural networks and extends these methods to quantum neural networks, showing their general applicability.

Findings

Harmonic neural networks outperform physics-informed neural networks in benchmarks.

Quantum harmonic neural networks demonstrate effective representation of harmonic functions.

The approach is broadly applicable to classical and quantum neural network architectures.

Abstract

Harmonic functions are abundant in nature, appearing in limiting cases of Maxwell's, Navier-Stokes equations, the heat and the wave equation. Consequently, there are many applications of harmonic functions from industrial process optimisation to robotic path planning and the calculation of first exit times of random walks. Despite their ubiquity and relevance, there have been few attempts to incorporate inductive biases towards harmonic functions in machine learning contexts. In this work, we demonstrate effective means of representing harmonic functions in neural networks and extend such results also to quantum neural networks to demonstrate the generality of our approach. We benchmark our approaches against (quantum) physics-informed neural networks, where we show favourable performance.